The Superization of Hochschild's Lemma and Restricted Lie-Rinehart Superalgebras

TL;DR

This paper generalizes Hochschild's lemma to the super setting, introduces restricted Lie-Rinehart superalgebras over fields of characteristic p>2, and constructs their universal enveloping algebras with key properties.

Contribution

It extends Hochschild's lemma to Lie-Rinehart superalgebras and defines the new concept of restricted Lie-Rinehart superalgebras, including their representations and universal enveloping algebras.

Findings

Established a superized Hochschild's lemma

Defined and explored properties of restricted Lie-Rinehart superalgebras

Constructed universal enveloping algebras with universal property

Abstract

The main goal of this paper is to introduce the notion of restricted Lie-Rinehart superalgebra over a field of characteristic $p>2$, motivated by a generalization of Hochschild's lemma to the super setting. We extend Schauenburg's proof of Hochschild's lemma to Lie-Rinehart superalgebras and we prove a superized version that serves as a foundation for our construction. Building upon this, we define restricted Lie-Rinehart superalgebras, investigate their representations, construct the semi-direct product with a restricted module, and provide several examples. Finally, we construct the corresponding universal enveloping algebra and show that this algebra satisfies the expected universal property.